Nonlocal operators in divergence form and existence theory for integrable data

2025/04/14 by Arcoya, David, Dipierro, Serena, Lippi, Edoardo Proietti +2 · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.09976

Abstract

We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to L¹(Ω) and to be suitably dominated. We also prove that the solution that we find converges, as s\nearrow 1, to a solution of the local counterpart problem, recovering the classical result as a limit case. This requires some nontrivial customized uniform estimates and representation formulas, given that the datum is only in L¹(Ω) and therefore the usual regularity theory cannot be leveraged to our benefit in this framework. The limit process uses a nonlocal operator, obtained as an affine transformation of a homogeneous kernel, which recovers, in the limit as s\nearrow 1, every classical operator in divergence form.

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